QR factorization

$A = QR$ — orthonormal columns ( $Q$ ) times an upper-triangular record of Gram-Schmidt ( $R$ ).

Running Gram-Schmidt on the columns of $A$ and packaging the orthonormal vectors as $Q$ 's columns and the projection coefficients as $R$ 's entries gives $A=QR$ , with $R$ upper triangular and positive diagonal entries — a record of the order Gram-Schmidt processed columns in, not an imposed condition.